A1 - u = ln x and then use integration by parts 5 Find R...

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Math 128: Assignment 1 due Wed, Sep 21 by 4:30 pm 1. Find dy/dx for the following (show your work): (a) y = x 2 sin( x ) + e x tan( x ) (b) sin 3 ( x ) + e sin( x ) Hint: Write e sin( x ) = f ( g ( x )), where f ( x ) = e x and g ( x ) = sin( x ); and use the Chain rule. (c) ln( x 2 + 5) = x 3 + y 2 (d) y = tan - 1 ( x 2 + 1) (e) y = ln( x 2 + x + 1) (f) y = x 3 sin( x ) Hnt: Use logarithmic differentiation. 2. Compute the following (you may use the Fundamental Theorem of Calculus): (a) Z 2 1 (6 x 2 - 4 x + 1) dx (b) Z 1 0 7 1 + x 2 dx (c) Let g ( x ) = Z x - 2 cos( πt ) tan( t ) dt . Compute g 0 (0). 3. Use the method of substitution to find the following: (a) Z 4 x 2 x 2 + 5 dx (b) Z (ln( x )) 3 x dx (c) Z π 0 x sin( x 2 ) dx (d) Z π/ 2 0 cos( x ) sin(sin x ) dx 4. Use integration by parts to find the following: (a) Z θ sin(2 θ ) (b) Z (ln x ) 2 x 3 dx

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(c) Z sin(ln x ) dx Hint: First make the substitution
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Unformatted text preview: u = ln x and then use integration by parts 5. Find R cos 2 ( x ) dx . Hint: Use integration by parts. You will end up with something in terms of R sin 2 ( x ) dx . From there, you can use the well-known identity cos 2 ( x ) + sin 2 ( x ) = 1 to replace the sin 2 ( x ) with something in terms of cos 2 ( x ). You will have an equation with your integral on both sides. You can rearrange and solve for the integral you are looking for. You might also want to look at Section 7.1, Example 6 in the text book....
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